Find Matrix Inverses; Solve Systems

The Matrix Analog of Division

In this lesson:

• Compute for matrices; test invertibility
• Solve via

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Explain :
2. Compute for matrices using the formula
3. Test invertibility: check
4. Solve via ; verify
5. Use technology for and larger systems

The Scalar Analogy for

Scalar: (when )

Matrix:

The identity matrix plays the role of 1: .

The matrix analog of division — when exists.

The Inverse:

The identity matrix: . Multiplying any matrix by leaves it unchanged.

The inverse satisfies:

Not every matrix has an inverse — we'll see when one exists.

Verifying That a Matrix Inverse Is Correct

Verify for :

The Determinant: Check Invertibility First

: is invertible  |  : is singular (no inverse)

Invertible vs. Singular: Two Examples

: → invertible

: → singular (no inverse)

Row 2 is row 1 — equations are dependent.

The 2×2 Inverse Formula: Swap-Negate-Divide

For with :

Mnemonic: ① Swap and   ② Negate and   ③ Divide by

Worked Example 1: Computing

: . Swap, negate, divide by 1:

Verify:

Worked Example 2: Singular Matrix

Find for

Singular matrix — no inverse. Stop here.

No unique solution exists for any system with this coefficient matrix.

Worked Example 3: Fractional Inverse

:

Verify: ✓

What a Singular Matrix Means for the System

When , the system has no unique solution:

• Inconsistent: no solution (lines are parallel — never meet)
• Dependent: infinitely many solutions (lines are identical)

Singular ≠ automatically "no solution." Check the specific system.

Common Error: Multiplying on the Wrong Side

Derivation — why goes on the LEFT:

Wrong: does not simplify to .

Matrix multiplication is not commutative — .

Solving a System with

Solve: , ;

Verify: ✓; ✓.

One Solves Many Systems

— computed once.

Check-In: Compute and Solve

Find and solve for ,

1. Compute
2. Apply the formula to find
3. Compute
4. Verify in both original equations

Technology for and Larger

The 2×2 formula doesn't extend to 3×3. Use technology:

TI-84: MATRIX menu → enter → compute

Desmos: matrix([[a,b,c],[d,e,f],[g,h,i]])^(-1)

The key skill is setup — entering and correctly. The calculator handles the arithmetic.

Unscaffolded: Full Matrix Inverse Solution

Solve algebraically: and

Full process: system → matrix form → determinant → → → verify.

No hints. Show every step.

What You Can Do Now

✓ Compute ; check before inverting

✓ Swap-negate-divide; verify

✓ Solve via

Determinant FIRST — zero means stop

Singular ≠ no solution — could be infinitely many

on the LEFT

Matrix Inverses Appear Throughout Applied Mathematics

The method you learned today is foundational across fields:

• Statistics: regression uses
• Graphics: transformation inverses restore original positions
• Physics: coupled equations in circuits and wave models
• AI: large matrix equations at every training step